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Super Black Hole from Cosmological Supergravity with a Massive hep-th/9801203 WATPHYS-TH-98/01 McGill/98-01 Super Black Hole from Cosmological Supergravity with a Massive Superparticle Marcia E. Knutt-Wehlau∗ Physics Department, McGill University, Montreal, PQ CANADA H3A 2T8 and R. B. Mann† Physics Department, University of Waterloo, Waterloo, ON CANADA N2L 3G1 ABSTRACT We describe in superspace a classical theory of two dimensional (1, 1) cosmological dilaton supergravity coupled to a massive superparticle. We give an exact non-trivial superspace solution for the compensator arXiv:hep-th/9801203v1 29 Jan 1998 superfield that describes the supergravity, and then use this solution to construct a model of a two-dimensional supersymmetric black hole. January 1998 ∗[email protected] †[email protected] 1 Introduction There are very few exact classical non-trivial solutions to supersymmetric field the- ories. As is elaborated on in [1], even more rare are exact superspace supergravity solutions. For classical supergravity theories, one looks for non-trivial solutions – those that cannot be reduced by infinitesimal supersymmetry transformations to purely bosonic solutions – using the method given in [2]. However, it is possible to sidestep this issue by examining classical supergravity problems in superspace [3]. A bona fide superspace supergravity solution – one which satisfies the con- straints – has non-trivial torsion, a supercovariant quantity, and as such its value ultimately remains unchanged under a suitable gauge transformation. Hence an exact superspace supergravity solution must necessarily be non-trivial in this sense. Approaching classical supergravity problems from the superspace viewpoint obviates the triviality question. It is this approach that we take in this paper. Following our previous motivation [1], we consider (1, 1) dilaton supergravity with a cosmological constant, coupled to a massive superparticle in (1 + 1) dimen- sions. The supergravity part of the theory is a supersymmetric generalization of the (1 + 1) dimensional “R=T” theory [4]. This theory has the unique feature that the dilaton superfield decouples from the classical equations of motion, so that super- matter induces superspace curvature, and the superspace curvature reacts back on the supermatter self-consistently. We obtain an exact solution for the supergravity compensator superfield that completely describes the supergravity in superconfor- mal gauge, and use this compensator to construct a model of a supersymmetric black hole. The outline of our paper is as follows. In section 2 we review bosonic cosmological dilaton gravity coupled to a massive particle. In section 3, we outline cosmological dilaton (1, 1) supergravity coupled to a massive superparticle. In section 4, we solve for the supergravity compensator in the presence of this superparticle, and in section 5, we discuss the construction of a super black hole model using this compensator. 2 Cosmological Dilaton Gravity Before describing the supergravity action we use, we briefly review the bosonic Lagrangian of a massive point particle interacting with cosmological dilaton gravity [5], as it is simpler than the supersymmetric case, and illustrates the basic ideas. The action for R = T theory is 1 2 1 2 S = SG + SM = d x √ g(ψR + ( ψ) ) κ M (2.1) 2κ Z − 2 ∇ − L 1 where the gravitational coupling κ =8πG. The action (2.1) ensures that the dilaton field ψ decouples from the classical equations of motion which, after some manipu- lation, are µ R = κTµ (2.2) 1 1 2 2 µψ νψ ( ψ) µ ν ψ + gµν ψ = κTµν (2.3) 2 ∇ ∇ − 2 ∇ −∇ ∇ ∇ 1 δ M λ λ λ σ where Tµν = √ g δgLµν is the stress-energy tensor and Rµν = ∂λΓµν ∂ν Γµλ ΓµσΓλν + λ σ − − − ΓλσΓµν is our convention for the Ricci tensor. We take the matter Lagrangian to be that of a point particle in a spacetime with non-zero cosmological constant dzα dzβ = √ gΛ+2m dτ g δ(2)(x z(τ)) (2.4) M s αβ L − − Z − dτ dτ − so that 1 1 dzα dzβ T = g Λ+ m dτ g g δ(2)(x z(τ)) (2.5) µν 2 µν √ g µα νβ dτ dτ − Z − is the relevant stress energy, with zµ(τ) being the worldline of the particle. Choosing a frame at rest with respect to the particle, the trace of the stress energy is µ ρ T = 2me− δ(x x ) + Λ (2.6) µ − − 0 2 2ρ + 2ρ 2 2 in conformal coordinates where ds = e dx dx− = e ( dt + dx )/4, with the − location of the particle at x = x0. The field equations (2.2) then become κ 2ρ ρ ρ′′(x)= Λe + Me δ(x x ) (2.7) − 8 − 0 with M =2πGm. Setting a2 = κ Λ , equation (2.7) has for Λ > 0 the solution 8 | | ρ = ln (cosh(a x x + b)) (2.8) − | − 0| M where sinh(b)= 2a . IfΛ < 0, the solution for M < 2a is ρ = ln (cos(a x x + b)) (2.9) − | − 0| M where sin(b)= 2a , or ρ = ln (sinh(b a x x )) (2.10) − − | − 0| M if M > 2a, where cosh(b)= 2a . 2 In Schwarzschild-type coordinates the metric in the Λ > 0 case (2.8) becomes κ dY 2 ds2 = ( ΛY 2 +2M Y + 1)dT 2 + (2.11) − − 2 | | κ ΛY 2 +2M Y +1 − 2 | | whereas for Λ < 0 the solutions (2.9) and (2.10) can respectively be transformed into κ dY 2 ds2 = ( Λ Y 2 +2M Y + 1)dT 2 + (2.12) − 2 | | | | κ Λ Y 2 +2M Y +1 2 | | | | and κ dY 2 ds2 = ( Λ Y 2 +2M Y 1)dT 2 + (2.13) − 2 | | | | − κ Λ Y 2 +2M Y 1 2 | | | | − The metric (2.13) is that of an anti de Sitter black hole with mass parameter M. This solution, along with (2.11) and (2.12), has been discussed previously in ref. [5]. 3 Cosmological (1, 1) Dilaton Supergravity We now extend the previous theory to a superspace formulation of (1, 1) dilaton supergravity in two dimensions with a cosmological constant, L. We use light-cone 1 1 0 + coordinates (x , x )= 2 (x x ) and (θ , θ−). In superconformal gauge, the action is given by ± 2 2 2 2S IC = d xd θ(D+ΦD Φ+4ΦD D+S 4e− L) (3.1) − − −κ Z − where Φ is the dilaton superfield, S is the scalar compensator superfield that com- pletely describes the supergravity in superconformal gauge, and the flat supersym- + metry covariant derivatives are given by D =(∂+ + iθ ∂ , ∂ + iθ−∂ ). We simply ± − list the results here, but details can be found in [1]. The equations of motion are Φ = 2S (3.2) − 2S D D+S = e− L (3.3) − from which it is clear that the dilaton decouples from the theory, and that we recover the superLiouville equation for the compensator, S. To obtain the component form of the superspace action (3.1), we identify the components of the superfields by theta expansion (dropping the fermionic fields), 1 + S = ρ + σθ θ− (3.4) −2 1 + Φ = ψ + ϕθ θ− (3.5) −2 3 and eliminate the auxiliary fields ϕ, σ via their equations of motion. This yields 1 I = d2x 4ψ∂ ∂ ρ + ∂ ψ∂ ψ 16L2e2ρ (3.6) C 2κ − − Z h i for the component action. This is equivalent to (2.1) (using (2.4) with m = 0) 32 2 provided Λ = κ L . Since L must be real, this implies that from the superspace action, only the− component action for anti de Sitter spacetimes is recovered. Alter- natively, inserting the superfield expansions (3.4,3.5) into eqs. (3.2) and (3.3) in the static case yields after some manipulation 2 2ρ κ 2ρ ρ′′(x)=4L e = Λe (3.7) − 8 which is equation (2.7) with M = 0. We consider now extending the action (3.1) to include a massive superparticle. We use z =(x, t, θ) as the coordinates of the superspace, and z0(t)=(x0(t), θ0(t)) as the coordinates of the superparticle. The technical details leading up to the action (3.8) below are the same as in [1], so we supply only the results here. The action for the superparticle in superconformal gauge is 2 1 4S 1 + + 1 I = 2m dtdxd θ g− e− (1 +x ˙ )+ iθ θ˙ (1 x˙ )+ iθ −θ˙ − P 2 0 0 0 2 − 0 0 0 Z n + h i h i 1 + ˙ 1 ˙ − + i 2 (1 +x ˙0)+ iθ0 θ0 D+G+ + i 2 (1 x˙0)+ iθ0−θ0 D G − − − h+ g i h i − + + + θ˙0 G+ + θ˙0 G + δ(x x0(t))δ(θ θ0 (t))δ(θ− θ0−(t)) (3.8) − 4 − − − where g is the einbein on the worldline of the superparticle, and G eSΓ . The α ≡ α general gauge superfield Γα necessarily appears in the Wess-Zumino type term in the massive superparticle action in order for consistent coupling of the flat superparticle to supergravity. Requiring that the supergravity constraints be satisfied introduces a constraint on the gauge field G, and we include this constraint in the supergravity action by means of a lagrange multiplier, λ. Consequently, the dilaton supergravity part of the action is affected and becomes 2 2 2 2S IC = d xd θ[D+ΦD Φ+4ΦD D+S 4e− L − − −κ Z − 2S 2S + κλe− (D+G + D G+ ie− )] (3.9) − − − From the sum of (3.8) and (3.9), we obtain for the equation of motion for S 2S D D+S(z) e− L − − κm 1 4S 1 + + 1 4 = dt′ g− e− (1 +x ˙ )+ iθ θ˙ (1 x˙ )+ iθ −θ˙ − δ (z z (t′)) 2 2 0 0 0 2 − 0 0 0 − 0 Z n h i h io κm 2 2S + + = √π e− δ(x x (t))δ(θ θ (t))δ(θ− θ −(t)) (3.10) 4 − 0 − 0 − 0 4 where √π2 = 1 1 x˙ 2 for a free particle.
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